The Overflow Problem Related To Draw-down In The Process Of Refracting Lévy

The refracted Lévy risk model and the risk model with Parisian delay have very impor-tant theoretical value and practical significance in the field of stochastic process theory and financial insurance.In this paper,we further investigate the exit problems associated with draw-down times for a refracted Lévy process.In the second chapter,we study the refracted Lévy risk model with draw-down times.In this risk process,replacing bankruptcy time by draw-down times,we obtain the expression of Laplace transformation at the first arrival time related to draw-down times.In the proof,we use the approximation method,and the results are expressed in the form of scale functions.Then two examples are given.We obtain the expression of PuI(?_?~+??)and the data list is given.In the third chapter,we discuss a draw-down refracted Lévy risk model with Parisian delay.The refracted Lévy risk model has two draw-down levels:?(U(?_?))and U(?_?),where?? is the draw-down time of the refracted Lévy risk process.Let X and Y be two differ-ent spectrally negative Lévy processes,and draw-down refracted Lévy process U initially agrees with the spectrally negative Lévy process X.When the process U falls below the level ?(U(T_?)),U becomes Y,until it reaches level U(?_?)again,and then becomes X.In this paper,we obtain the Laplace transform expression related to the first arrival time of the refracted Lévy process,and study the Laplace transform expression of the exit problems with Parisian delay.Finally,we give two examples with linear draw-down functions.